SECTION III. MENSURATION OF SOLIDS FOUNDED BY PLANE SURFACES. DEFINITIONS. ART. 36. Mensuration of Solids is divided into two parts: I. The mensuration of the surfaces of solids; II. The mensuration of their solidities. (Art. 1.) It has already been shown (Art. 2, Def. IV.) that the unit of measure for plane surfaces is a square, whose side is a foot, a yard, or any other fixed quantity. 1. A Prism is a solid whose ends are parallel, similar and equal, and the sides, connecting these are parallelograms. A prism takes particular names according to the figure of its base, whether triangular, square, rectangular, pentagonal, &c. The parallel planes are sometimes called bases or ends. The perpendicular distance between the bases is called the height of the prism. 2. A Parallelopiped is a prism bounded by six quadrilateral planes, every opposite two of which are equal and parallel. 3. A Cube is a right prism, bounded by six equal square faces, of which any two, opposite to each other, are parallel. 4. A Pyramid is a solid whose base is any plane figure, and whose sides are triangles, having all their vertices meeting together in a point above the base, called the vertex of the pyramid. The perpendicular distance from the vertex to the plane of the base is the height of the pyramid; as, DE. The slant height of a pyramid is a line drawn from the vertex to the middle of one of the sides of the base. A pyramid, like the prism, takes particular names from the figure of the base, according as it is square, triangular or polygonal. 1 B 5. A Frustrum or Trunk of a pyramid is a portion of the solid that remains after any part has been cut off parallel to the base. The height of the frustrum is a line drawn through the centre of the pyramid from the centres of the two parallel planes. The slant height is a line passing on the surface of the frustrum through the middle of either of its sides. F D E A F C C 8. Surface is the exterior part of any thing that has length and breadth, the limits that terminate a solid. Convex and lateral surface, are sometimes used synonymously in Mathematics. PROBLEM I. To find the Lateral Surface of a Right Prism. ART. 37. Rule.- Multiply the perimeter of the base into the altitude, and the product will be the convex surface. When the entire surface of the prism is required, add to the convex surface the area of the bases. Hence, the superfices of any solid, bounded by planes, is equal to the sum of the areas of all its sides. It is manifest that each of the sides of the prism is a parallelogram, whose area is equal to the product of the length into the breadth. Now, since the breadth is only one side of the base, therefore the sum of all the breadths is equal to the perimeter of the base. Ex. 1. What is the entire surface of a regular prism, whose base is a regular pentagon, each side of which is 20 feet, and whose altitude is 50 feet ? OPERATION. 20×5=100=perimeter. 5000 sq. ft. which is the convex surface. We have for the area of the end (by Art. 13) 20xtabular number, or 400×1.720477=688.1908. Then, 688.1908=area of one end. Ex. 2. Required the lateral surface of a prism whose base is a regular hexagon, and whose sides are each 2 feet 3 inches, the height being 11 feet? Ans. 216 sq. ft. Ex. 3. What is the entire surface of a triangular prism, whose base is an equilateral triangle, having each of its sides equal to 18 inches, and altitude 20 feet? Ans.' 91.949 sq. ft. Ex. 4. What is the surface of a regular Heptagonal prism, each side of whose base is 16 and altitude 15 feet ? Ans. 1680 sq. ft. PROBLEM II. To find the Solidity of a Prism. ART. 38. Rule.-Multiply the area of the base by the perpendicular height, and the product will be the solid con tents. NOTE. The above rule holds true whatever the figure of the base may be, whether right or oblique parallelopipedons, cubes, &c. As surfaces are measured by comparing them with a right parallelogram, so solids are measured by comparing them with a right parallelopipedon. If the base of a right parallelopipedon be given, it is manifest that the number of cubic feet contained in one foot of the height, is equal to the number of square feet in the base. And if the solid be of any other height whatever, instead of one foot, the contents will be in the same ratio. Let us illustrate this by an example: Let ABCD be the base of a right parallelopipedon, and suppose AB & BC each =4 feet. Then, the number of square feet in the base ABCD will be equal to 4x4=16 square feet. D Now, it is manifest that a parallelopipedon 1 foot in height contains 16 cubic feet; and were it 2 feet in height, it would A contain 32 cubic feet, &c. B That is, the contents of any parallelopiped is found by multiplying the area of the base by the altitude of that solid. So, on the other hand, if the solid contents of a cubic body be given, the length of the edges may be found by extracting the cube root of the given solid. Ex. 1. What is the solidity of a wall 28 feet long, 12 feet high, and 3 feet 4 inches thick ? |